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bonanova
Posted 11 March 2013 - 09:08 AM
Warm-up problem:
Bisect the angles of a triangle.
Describe the point(s) where each bisector first intersects one of the others.
Now try this:
Trisect the angles of a triangle.
Describe the points where each trisector first intersects one of the others.
Junior Member
Posted 11 March 2013 - 12:38 PM
Do we have to name that point?
bonanova
Posted 11 March 2013 - 03:19 PM
Do we have to name that point?
Spoiler for
You're right about the bisector case.
But for the trisector case there is more than one point.
In fact there are three places where a trisector of one angle first intersects a trisector of one of the other angles.
And there is something special about those three points.
Junior Member
Posted 12 March 2013 - 09:49 AM
Do we have to name that point?
Spoiler for
You're right about the bisector case.
But for the trisector case there is more than one point.
In fact there are three places where a trisector of one angle first intersects a trisector of one of the other angles.
And there is something special about those three points.
bonanova
Posted 14 March 2013 - 04:38 PM
Do we have to name that point?
Spoiler for
You're right about the bisector case.
But for the trisector case there is more than one point.
In fact there are three places where a trisector of one angle first intersects a trisector of one of the other angles.
And there is something special about those three points.
Spoiler for i think
Actually they are not collinear.
That being the case, they form a triangle.
So the OP really asks: what is special about the triangle formed by these three points?
Newbie
Posted 14 March 2013 - 09:47 PM Best Answer
bonanova
Posted 15 March 2013 - 05:25 AM
Spoiler for It seems...
Correct. Good job.
I wonder if there is a proof of this that is not overly complex?
Edit:
Well, No. I just found the proof, and it's not beautiful for its simplicity.
You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.
"Do not try this at home." 
Edited by bonanova, 15 March 2013 - 06:23 AM.
Comment on proof
Junior Member
Posted 18 March 2013 - 01:01 PM
Spoiler for It seems...Correct. Good job.
I wonder if there is a proof of this that is not overly complex?
Edit:
Well, No. I just found the proof, and it's not beautiful for its simplicity.
You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.
"Do not try this at home."
oh i am so sorry .....i mistook it for the eulers line....my bad... 
Advanced Member
Posted 24 March 2013 - 03:09 AM
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